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Full text of "Treatise On Analysis Vol-Ii"

10    CONVOLUTION OF TWO FUNCTIONS        287

(defined almost everywhere, relative to /?) w locally $-integrable.  When this
condition is satisfied, the function

h(x) =    g(s~~lx)f(s) d/?(s),
defined almost everywhere relative to /?, is locally f$-integrable, and

This is a particular case of (14.8.1).

When the conditions of (14.10.1) are satisfied, the functions / and g are
said to be convolvable (with respect to /?). Any function which is equal almost
everywhere (with respect to /?) to the function h above is called a convolution
off and g (with respect to /!) and is written/* g (or/* ^g, where it is necessary
to bring j8 into the notation). Thus for almost all x we have

(14.10.2)              (f*g)(x)
and likewise, using (14.8.4),

(14.10.3)             (f*g)(x) = f/(xs-'

almost everywhere with respect to /?.

When one of the functions equal to h almost everywhere is continuous on
G, we adopt the same convention as in (14.8) and call this function the con-
volution of/ and g.

In particular, when/* g is continuous, we have

4

(14.10.4)                     (/ * g)(e) = I f(s)g(s" ') dfls).

It should be remarked that the property of /and g of being convolvable
does not depend on the choice of left Haar measure /?, but their con-
volution/*^ does. If /? is replaced by a/?, where a > 0, then we have
f+ęg = a-f.fg

When G is discrete, to say that /and g are convolvable signifies that the
family (g(s~*x)f(s))$eQ is absolutely summable (5.3.3) for all x e G, and we
have

seG

if the Haar measure j? is such that ft({e}) = 1.ft-invariant distance on G. Then use (c).)